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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgeev (f08na)

## Purpose

nag_lapack_dgeev (f08na) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an $n$ by $n$ real nonsymmetric matrix $A$.

## Syntax

[a, wr, wi, vl, vr, info] = f08na(jobvl, jobvr, a, 'n', n)
[a, wr, wi, vl, vr, info] = nag_lapack_dgeev(jobvl, jobvr, a, 'n', n)

## Description

The right eigenvector ${v}_{j}$ of $A$ satisfies
 $A vj = λj vj$
where ${\lambda }_{j}$ is the $j$th eigenvalue of $A$. The left eigenvector ${u}_{j}$ of $A$ satisfies
 $ujH A = λj ujH$
where ${u}_{j}^{\mathrm{H}}$ denotes the conjugate transpose of ${u}_{j}$.
The matrix $A$ is first reduced to upper Hessenberg form by means of orthogonal similarity transformations, and the $QR$ algorithm is then used to further reduce the matrix to upper quasi-triangular Schur form, $T$, with $1$ by $1$ and $2$ by $2$ blocks on the main diagonal. The eigenvalues are computed from $T$, the $2$ by $2$ blocks corresponding to complex conjugate pairs and, optionally, the eigenvectors of $T$ are computed and backtransformed to the eigenvectors of $A$.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{jobvl}$ – string (length ≥ 1)
If ${\mathbf{jobvl}}=\text{'N'}$, the left eigenvectors of $A$ are not computed.
If ${\mathbf{jobvl}}=\text{'V'}$, the left eigenvectors of $A$ are computed.
Constraint: ${\mathbf{jobvl}}=\text{'N'}$ or $\text{'V'}$.
2:     $\mathrm{jobvr}$ – string (length ≥ 1)
If ${\mathbf{jobvr}}=\text{'N'}$, the right eigenvectors of $A$ are not computed.
If ${\mathbf{jobvr}}=\text{'V'}$, the right eigenvectors of $A$ are computed.
Constraint: ${\mathbf{jobvr}}=\text{'N'}$ or $\text{'V'}$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ matrix $A$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a. (An error is raised if these dimensions are not equal.)
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
2:     $\mathrm{wr}\left(:\right)$ – double array
3:     $\mathrm{wi}\left(:\right)$ – double array
The dimension of the arrays wr and wi will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.
4:     $\mathrm{vl}\left(\mathit{ldvl},:\right)$ – double array
The first dimension, $\mathit{ldvl}$, of the array vl will be
• if ${\mathbf{jobvl}}=\text{'V'}$, $\mathit{ldvl}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise $\mathit{ldvl}=1$.
The second dimension of the array vl will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvl}}=\text{'V'}$ and $1$ otherwise.
If ${\mathbf{jobvl}}=\text{'V'}$, the left eigenvectors ${u}_{j}$ are stored one after another in the columns of vl, in the same order as their corresponding eigenvalues. If the $j$th eigenvalue is real, then ${u}_{j}={\mathbf{vl}}\left(:,j\right)$, the $j$th column of vl. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then ${u}_{j}={\mathbf{vl}}\left(:,j\right)+i×{\mathbf{vl}}\left(:,j+1\right)$ and ${u}_{j+1}={\mathbf{vl}}\left(:,j\right)-i×{\mathbf{vl}}\left(:,j+1\right)$.
If ${\mathbf{jobvl}}=\text{'N'}$, vl is not referenced.
5:     $\mathrm{vr}\left(\mathit{ldvr},:\right)$ – double array
The first dimension, $\mathit{ldvr}$, of the array vr will be
• if ${\mathbf{jobvr}}=\text{'V'}$, $\mathit{ldvr}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise $\mathit{ldvr}=1$.
The second dimension of the array vr will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvr}}=\text{'V'}$ and $1$ otherwise.
If ${\mathbf{jobvr}}=\text{'V'}$, the right eigenvectors ${v}_{j}$ are stored one after another in the columns of vr, in the same order as their corresponding eigenvalues. If the $j$th eigenvalue is real, then ${v}_{j}={\mathbf{vr}}\left(:,j\right)$, the $j$th column of vr. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then ${v}_{j}={\mathbf{vr}}\left(:,j\right)+i×{\mathbf{vr}}\left(:,j+1\right)$ and ${v}_{j+1}={\mathbf{vr}}\left(:,j\right)-i×{\mathbf{vr}}\left(:,j+1\right)$.
If ${\mathbf{jobvr}}=\text{'N'}$, vr is not referenced.
6:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  ${\mathbf{info}}>0$
The $QR$ algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements $_$ to n of wr and wi contain eigenvalues which have converged.

## Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real.
The total number of floating-point operations is proportional to ${n}^{3}$.
The complex analogue of this function is nag_lapack_zgeev (f08nn).

## Example

This example finds all the eigenvalues and right eigenvectors of the matrix
 $A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 .$
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
```function f08na_example

fprintf('f08na example results\n\n');

% Matrix A
n = 4;
a = [0.35,  0.45, -0.14, -0.17;
0.09,  0.07, -0.54,  0.35;
-0.44, -0.33, -0.03,  0.17;
0.25, -0.32, -0.13,  0.11];

% Eigenvalues and right eigenvectors of A
jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
[~, wr, wi, ~, vr, info] = f08na( ...
jobvl, jobvr, a);

fprintf('Index  Eigenvalue                Eigenvector\n');
k = 1;
conjugating = false;
for j = 1:n
fprintf('%3d', j);
if wi(j)==0 & ~conjugating
fprintf('  %12.4e%15s',wr(j),' ');
for l = 1:n
if (l>1)
fprintf('%32s', ' ');
end
fprintf('%12.4e\n',vr(l,k));
end
k = k + 1;
else
if conjugating
pl = '-';
mi = '+';
else
pl = '+';
mi = '-';
end
fprintf('  %12.4e %s %10.4ei ', wr(j), pl, abs(wi(j)));
for l = 1:n
if (l>1)
fprintf('%32s', ' ');
end
if vr(l,k+1)>0
fprintf('%12.4e %s %10.4ei\n', vr(l,k), pl, vr(l,k+1));
else
fprintf('%12.4e %s %10.4ei\n', vr(l,k), mi, abs(vr(l,k+1)));
end
end
if conjugating
k = k + 2;
end
conjugating = ~conjugating;
end
fprintf('\n');
end

```
```f08na example results

Index  Eigenvalue                Eigenvector
1    7.9948e-01                -6.5509e-01
-5.2363e-01
5.3622e-01
-9.5607e-02

2   -9.9412e-02 + 4.0079e-01i  -1.9330e-01 + 2.5463e-01i
2.5186e-01 - 5.2240e-01i
9.7182e-02 - 3.0838e-01i
6.7595e-01 - 0.0000e+00i

3   -9.9412e-02 - 4.0079e-01i  -1.9330e-01 - 2.5463e-01i
2.5186e-01 + 5.2240e-01i
9.7182e-02 + 3.0838e-01i
6.7595e-01 + 0.0000e+00i

4   -1.0066e-01                 1.2533e-01
3.3202e-01
5.9384e-01
7.2209e-01

```